The Constrained Minimun-norm Least Squares Solutions Of A Class Of Matrix Equation

The problem of solving linear matrix equation is one of the important researchfields in the numerical linear algebra. It has many applications in biology, electricity,molecular spectroscopy, vibration theory, finite elements, structural design, solid me-chanics, parameter identification, automatic control theory, linear optimal control andso on.This thesis is mainly concerned with the problems about how to get the con-strained minimun–norm least squares solution for a class of matrix equation. Theseproblems are described as follows:Problem I Given A∈R^m×n, B∈R^n×p, C∈R^m×k, D∈R^k×s andE∈R^m×s, find X∈S₁ (?) R^n×n, Y∈S₂ (?) R^k×k such thatwhere·is Frobenius norm, the notation in the following is the same as in ProblemI.Problem II Let SE denote the solution set of Problem I, find [X,Y ]∈SE, suchthatIn this thesis , we mainly study Problem I,II when S1,S2 are Toeplitz matrix, generatorconstraint Toeplitz matrix, cycle matrix, generator constraint cycle matrix, Hankelmatrix, generator constraint Hankel matrix, respectively.The conventional matrix decomposition methods have been used to find the leastsquares solutions of the above mentioned inconsistent matrix equations over givenmatrix set in many references, and the general expressions of these solutions wereobtained. But it is di?cult to determine the solution of Problem I, II by utilizing theseexpressions due to the fact that the orthogonal invariance of Frobenius norm does nothold for the general nonsingular matrices. A series recent references about two matrixdecomposition methods are applied simultaneously to overcome this di?culty skillfully,and the expressions of the solutions for Problem I, II are also obtained. However, Itseems that there are some di?culties in finding the least squares constrained (forexample, Toeplitz) solution of matrix equation AXB + CY D = E with the leastnorm by use of the conventional matrix decomposition methods. So we use the matrixKronecker Product, vec operation and Moore–Penrose inverse with the basis matrix of constraint matrix to transform the constraint problem in the problem I into theunconstraint problem, and obtain the general form of the least squares solutions andthe corresponding the minimal–norm least squares solution.